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(4.1a) Set

A

and partitioned

(4.1b) lower approximation

(4.1c) upper approximation

space

FIGURE 4.1: Upper and lower approximation of a set A in the partitioned space of the

form (j, j + 1]
2

Topology of Rough sets

To study the topological properties of rough set, we define a partition topology on X, based

on partitions induced by the equivalence relation∼
X,ϕ
. The equivalence classes x
/∼
form

the basis for the topology τ (Steen and Seebach, 1995).

Let A denote the closure of a set A. Since the topology is a partition topology, to find

the closure A of a subset A⊂X, we have to consider all of the closed sets containing the

set A and then select the smallest closed set. The interior of a set A (denoted A) is the

largest open set that is contained in A. Open sets and their corresponding closed sets in

the topology are:

•X is an open set, φ is the corresponding closed set.

•φ is a closed set, X is the corresponding closed set.

•x
i
/
∼
is an open set⇒X\x
i
/
∼
= x
i
/
∼
, is a closed set.

where i = 1, . . . , n, and n is the total number of equivalence classes in the partition topology

and\is the set difference. Recall that the union of any finite or infinite number of open

sets are open sets and the intersection of any finite number of open sets is an open set. For

a set A⊆X and the partition topology τ , we have:

A = A
−

(4.18)

A = A
−
(4.19)

where A, A,A
−
and A
−
are closure, interior, upper and lower approximation of A, respec-

tively. Now it is clear that we could define properties of a rough set in the language of

topology. Next, define π−boundary of A as: A
b
= A\A, where A and A are closure and

interior of A, (Lashin, Kozae, Abo Khadra, and Medhat, 2005). A set A is said to be rough,

if A
b

6= φ, otherwise it is an exact set. Generally, for a given topology τ and A⊆X, we

have:

A = A.

•A is totally definable, if A is an exact set, A =

•A is internally definable, if A = A, A 6= A.

•A is externally definable, if A 6= A, A = A.

•A is undefinable, if A 6=

A and A 6= A.

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